THE F -DIFFERENT AND A CANONICAL BUNDLE FORMULA OMPROKASH DAS AND KARL SCHWEDE Abstract. We study the structure of Frobenius splittings (and generalizations thereof) induced on compatible subvarieties W ⊆ X. In particular, if the compatible splitting comes from a compatible splitting of a divisor on some birational model E ⊆ X0 −! X (ie, this is a log canonical center), then we show that the divisor corresponding to the splitting on W is bounded below by the divisorial part of the different as studied by Kawamata, Shokurov, Ambro and others. We also show that difference between the divisor associated to the splitting and the divisorial part of the different is largely governed by the (non-)Frobenius splitting of fibers of E −! W . In doing this analysis, we recover an F -canonical bundle formula by reinterpretting techniques common in the theory of Frobenius splittings. 1. Introduction This paper explores several questions that show up naturally in the study of Frobenius splittings, and also appear naturally as one compares the theory of F -singularities to the theory of the singularities of the minimal model program. Question 1.1 (Frobenius splitting perspective). Suppose X is a Frobenius split variety by a splitting φ and W is compatibly φ-split with induced splitting φW . Now, φW corresponds to a divisor D. ◦ What divisor is it? ◦ Where does it come from? ◦ How does it vary as the characteristic varies? This question turns out to be much more interesting when W has codimension at least 2 in X. The case where W has codimension 1 has already been largely answered in [Das15]. As mentioned, a very closely related question also appears in the comparison of F - singularities with the singularities of the minimal model program. Let us sketch the situation under some simplifying assumptions. Suppose that (X; ∆) is a proper log canonical pair and that W ⊆ X is a normal (or minimal for simplicity) log canonical center. Suppose the codimension of W in X is at least 2. Then one would like there to be divisors DiffW and MW on W such that (1.1.1) (KX + ∆)jW ∼Q KW + DiffW +MW where (a) DiffW is effective and (W; DiffW ) is log canonical (or in the minimal case, even KLT), and (b) MW is semi-ample. 2010 Mathematics Subject Classification. 14F18, 13A35, 14B05, 14D99, 14E99. The first named author was supported by NSF FRG Grant DMS #1265285, NSF Grant DMS #1300750 and Simons Grant Award #256202. The second named author was supported in part by the NSF FRG Grant DMS #1265261/1501115, NSF CAREER Grant DMS #1252860/1501102 and a Sloan Fellowship. 1 2 OMPROKASH DAS AND SCHWEDE While such objects do not exist in general, some weak versions of them do exist, at least on birational models of W , see [Kaw97],[Kaw98], [Amb04, Theorem 0.2], [Hac14], and [PS09, Conjecture 7.13]. In characteristic p > 0, the F -singularities analog of log canonical centers are F -pure centers (which are a generalization of compatibly split subvarieties). For instance, every log canonical center becomes an F -pure center at least in an ambient F -pure pair. Relevant for us, in characteristic p > 0, if (X; ∆) is sharply F -pure and the index of KX + ∆ is not divisible by p, then if W ⊆ X is a normal F -pure center, there is a canonically determined effective divisor ∆W;F - diff on W such that (1.1.2) (KX + ∆)jW ∼Q KW + ∆W;F - diff where (a')( W; ∆W;F - diff ) is F -pure. Note in this case there doesn't initially appear to be a moduli part, there is a single effective divisor. Another way to phrase our initial question is: Question 1.2 (Singularities of the MMP perspective). Suppose that (X; ∆) is a sharply F -pure pair and W ⊆ X is a normal F -pure center. Consider ∆W;F - diff the F -different on W . ◦ How does ∆F - diff compare with DiffW and MW ? ◦ Where does it come from? ◦ How does it vary as the characteristic varies? Our first result is: Theorem A. Suppose that (X; ∆) is a sharply F -pure (and hence log canonical) pair and W ⊆ X is a log canonical center (and hence an F -pure center). Let ν : W N −! W be the normalization of W and let DiffW N and ∆W N;F - diff be the different and F -different respectively. Then DiffW N ≤ ∆W N;F - diff . It then becomes very natural to study the difference ∆W N;F - diff − DiffW which should be viewed as some characteristic p > 0 analog of the moduli part of the different, already we know it is an effective divisor. In characteristic zero, the moduli part of the different comes from analyzing a family. Consider the following situation. Let (X0; ∆0) −!π (X; ∆) be 0 ∗ 0 a birational model of X with KX0 + ∆ = π (KX + ∆), such that ∆ ≥ 0 and such that E is a Q-Cartier prime divisor with discrepancy −1 such that π(E) = W . Then πE : E −! W can be viewed as a family, indeed a flat family if W is a curve. Note E has coefficient 1 in 0 0 ∆ , and so since KX0 + ∆ is pulled back from the base, if ∆E is the ordinary different of 0 0 0 0 KX +∆ on E (in particular, (KX +∆ )jE = KE +∆E), then KE +∆E ∼Q,πE 0. It is then natural to try to study the moduli part of the different via a canonical bundle formula, i.e., ∗ find a divisor ∆W ≥ 0 such that πE(KW + ∆W ) ∼Q KE + ∆E. It turns out that using a simple method coming from the origins of Frobenius splitting theory, we obtain a canonical bundle formula in characteristic p > 0. Theorem B. [Theorem 5.2, Corollary 5.12] Suppose that π : E −! W is a proper map between normal F -finite integral schemes with π∗OE = OW . Suppose also that ∆ ≥ 0 is a Q- e divisor on E such that (p −1)(KE +∆) is linearly equivalent to the pullback of some Cartier divisor on W . Suppose that the generic fiber of (E; ∆) over W is Frobenius split. Then ∗ there exists a canonically determined Q-divisor ∆W ≥ 0 on W such that π (KW + ∆W ) ∼Q KX + ∆. THE F -DIFFERENT AND A CANONICAL BUNDLE FORMULA 3 Furthermore, we can describe the support of ∆W as follows. Assume that W is 1- dimensional (which we may certainly do if we just care about the support of ∆W ) and that ∆vert is the vertical part of ∆. If additionally the fibers of π are geometrically normal, ∗ then π ∆W − ∆vert is nonzero precisely over those points t 2 W where the fiber (Et; ∆t) is not Frobenius split. Indeed, we also obtain versions of the above result which hold when ∆ is not necessarily effective, see Corollary 5.4. Remark 1.3. For those coming from the Frobenius splitting perspective, this result can be specialized follows. Suppose X is Frobenius split and the splitting extends to a resolution of singularities X0 −! X, and there is a compatibly split divisor E ⊆ X0 mapping to the necessarily compatibly split W ⊆ X. Then the splitting on W , and the corresponding divisor on W , is governed by the fibers of E −! W which are not Frobenius split (in a way somewhat compatible with the splitting of X0) at least assuming the fibers are sufficiently nice. Note that less well behaved fibers can be handled too, see Section 5). In characteristic p > 0, if X is 3-dimensional, (X; ∆) is sharply F -pure and W is a 1-dimensional minimal log canonical center, then we can obtain E −! W satisfying the first part of the above theorem via the MMP [HX15, Bir13, BW14]. In some cases, it is also possible to reduce to the case of integral fibers by employing base change, semi-stable reduction and results on M0;n (see [DH15, Theorem 4.8]). This gives us a precise way to describe the F -different, see Algorithm 5.13 for details. We also tackle the question of how the F -different behaves as the characteristic varies. To do this, we first need to assume the weak ordinarity conjecture (which implies that log canonical singularities become F -pure after reduction to characteristic p > 0 [MS11, Tak13]). Theorem C. [Theorem 6.2] Let (X; ∆ ≥ 0) be a normal pair in characteristic zero with e (p − 1)(KX + ∆) Cartier for some e > 0, and W a normal LC-center of (X; ∆). Assume 1 that the b-divisor K + ∆div descends to W ; and in particular KW + ∆W;div is Q-Cartier. We consider the behavior of (Xp; ∆p) after reduction to characteristic p 0. Assume the weak ordinarity conjecture. Let Q 2 W be a point which is not the generic point of W . Then there exist infinitely many primes p > 0 such that if ∆Wp;F ≥ ∆W;div;p is the F -different of (Xp; ∆p) along Wp, then ∆Wp;F = ∆W;div;p near Q. In other words, Q is not contained in Supp(∆Wp;F − ∆W;div;p). Thanks: The authors would like to thank Christopher Hacon and Shunsuke Takagi for many useful discussions. We would also like to thank Christopher Hacon for useful comments on a previous draft. We also thank David Speyer and Chenyang Xu for stimulating discussions with the second author where Example 3.2 was first worked out. 2. Preliminaries Convention 2.1. Throughout this article, all schemes are assumed to be separated and excellent. All schemes in characteristic zero are of essentially finite type over a field. All schemes in characteristic p > 0 are assumed to be F -finite.